The parameter \(\rho \)  is increased or decreased by a factor \(\nu _\rho > 1\)  to ensure that the magnitudes of \(p^k\)  and \(d^k\)  are within a factor \(\chi > 1\)  of each other, similar to [1]}.
However, the number of times \(\rho \)  is allowed to decrease is bounded by some integer \(\Psi > 0\) , and moreover, it is not allowed to increase after \(\rho = \omega \theta \) .
Therefore, \(\rho \)  will not be adjusted an infinite number of times.
The proximal weight \(\tau _z\)  is fixed at \(\kappa _z \rho \) , where \(\kappa _z\)  is selected to ensure that \(\eta _z > 0\)  whenever \(\rho \)  is also sufficiently large.
In particular, the proof of Theorem REF  shows that whenever \(\rho /\theta = \omega > 32\) , then \(\kappa _z = 1/32\)  suffices.
Finally, since Theorem REF  ensures that the termination condition in line REF  will be met after a finite number of iterations (e.g., when \(\eta _x, \eta _z > 0\) ), the parameter \(\theta \)  will not be increased an infinite number of times in line REF .
Therefore, Algorithm REF  is guaranteed to converge to an \(\epsilon \) -approximate stationary point of the original problem (REF ) after \(O(\epsilon ^{-4})\)  iterations.
